A shifted cyclic quadratic ratio bound with a triple-product denominator

Proved

WorkbookSource.base_36731

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $x,y,z$ be positive reals satisfy $x+y+z=3$ , prove that $\frac{x}{z+y^2}+\frac{y}{x+z^2}+\frac{z}{x^2+y} \geq \frac{15}{9+xyz}$

Source: InternLM Lean-Workbook, record lean_workbook_36731 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36731 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (x / (z + y ^ 2) + y / (x + z ^ 2) + z / (x ^ 2 + y)) ≥ 15 / (9 + x * y * z)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36731; Apache-2.0

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